Invertible Toeplitz products
Let \(P^{+}\) denote the Hardy projection from \(L^{2}(\partial \mathbb D)\) onto the Hardy space \(H^{2}\). Let \(h\in L^{2}(\partial \mathbb D)\), define the Toeplitz operator \(T_{h}\) on \(H^{2}\) by the formula \(T_{h}p=P^{+}(hp)\) for polynomials \(p\). This operator is bounded if and only if \(h\) is bounded on the unit circle \(\partial \mathbb D\). \textit{D. Sarason} [Oper. Theory, Adv. Appl. 41, 485-496 (1989; Zbl 0678.46019) and ibid. 48, 333-347 (1990; Zbl 0744.46042)] found examples of \(f\) and \(g\) in \(H^{2}\) such that the product \(T_{f}T_{\bar{g}}\) is actually a bounded operator on \(H^{2}\) though neither \(T_{f}\) nor \(T_{\bar{g}}\) is bounded. In the paper under review, the authors find simple necessary an sufficient condition for boundedness and (simultaneously) invertibility of the product \(T_{f}T_{\bar{g}}\), \(f\), \(g\in H^{2}\). They give a criterion for boundedness and Fredholm property for these operators. Similar results are obtained for the Bergman space.
scientific article; zbMATH DE number 1888341
scientific article